Order Relations

Ordered Structure of the Integers

The integers are not merely a collection of numbers equipped with arithmetic operations. They also possess an order structure. Given two integers $a$ and $b$, one can determine whether $a$ is less than, equal to, or greater than $b$.

The order relation on the integers is written using the symbols

If

we say that $a$ is less than $b$. Equivalently, $b$ is greater than $a$.

The relation

means either

or

Similarly,

means either

or

The integers therefore form a linearly ordered set:

Every pair of integers can be compared uniquely.

Trichotomy Law

The most basic property of order is the trichotomy law.

For any integers $a$ and $b$, exactly one of the following statements holds:

Two distinct possibilities cannot occur simultaneously. For example, it is impossible for both

and

to hold.

The trichotomy law ensures that the integers are completely ordered.

Transitivity

The order relation is transitive. If

and

then

For example,

and

imply

Transitivity allows chains of inequalities to be combined into larger comparisons.

Compatibility with Addition

Order interacts naturally with arithmetic operations. If

then adding the same integer $c$ to both sides preserves the inequality:

For example,

implies

that is,

This property allows equations and inequalities to be manipulated algebraically.

Conversely, subtracting the same quantity from both sides also preserves order:

Compatibility with Multiplication

Multiplication behaves differently depending on the sign of the multiplier.

If

and

then

For example,

implies

so

However, multiplication by a negative integer reverses the inequality. If

and

then

For example,

implies

since

This reversal is one of the characteristic features of inequalities.

Positive and Negative Integers

The order relation divides the integers into three classes.

An integer $a$ is positive if

It is negative if

The integer $0$ is neither positive nor negative.

Every nonzero integer is therefore either positive or negative, but not both.

The sign of a product follows from the order structure:

• the product of two positive integers is positive,
• the product of two negative integers is positive,
• the product of integers with opposite signs is negative.

These facts are fundamental throughout arithmetic and algebra.

Bounds

Let $S$ be a set of integers.

An integer $u$ is called an upper bound for $S$ if

for every $a\in S$.

Similarly, an integer $l$ is called a lower bound if

for every $a\in S$.

For example, the set

has upper bound $4$ and lower bound $1$.

Finite sets of integers always possess both upper and lower bounds.

Well-Ordering Principle

One of the most important properties of the natural numbers is the well-ordering principle.

Every nonempty subset of $\mathbb{N}$ possesses a least element.

For example, the set

has least element $5$.

Even infinite subsets satisfy this property. The set of even positive integers

has least element $2$.

The well-ordering principle is equivalent to mathematical induction and plays a central role in proofs throughout number theory.

Order and Number Theory

Order relations appear constantly in arithmetic arguments. Statements about divisibility, primes, factorization, and congruences frequently depend on inequalities.

For example, if $p$ is a prime number and

one often studies the relative sizes of the quantities involved. Estimates for arithmetic functions, prime counting functions, and Diophantine equations all rely heavily on ordered structure.

Thus the integers combine two fundamental features:

1. arithmetic operations,
2. linear order.

The interaction between these structures is one of the foundations of number theory.